Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/ExProp7_Luc06_FR)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by NaTT @ termCOMP 2023)

1 Dependency Pair Transformation

1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

• The 1^st component contains the pair

  activate#(n__f(X))	→	activate#(X)	(14)
  activate#(n__s(X))	→	activate#(X)	(13)

  1.1.1 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

  [0#]	=	0
  [s(x1)]	=	0
  [activate(x1)]	=	0
  [activate#(x1)]	=	x1 + 0
  [p#(x1)]	=	0
  [f(x1)]	=	0
  [n__s(x1)]	=	x1 + 1
  [p(x1)]	=	0
  [0]	=	0
  [s#(x1)]	=	0
  [n__f(x1)]	=	x1 + 1
  [f#(x1)]	=	0
  [n__0]	=	0
  [cons(x1, x2)]	=	0

  having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the pairs

  activate#(n__f(X))	→	activate#(X)	(14)
  activate#(n__s(X))	→	activate#(X)	(13)

  could be deleted.

  1.1.1.1 Dependency Graph Processor

  The dependency pairs are split into 0 components.

• The 2^nd component contains the pair

  f#(s(0))

  →

  f#(p(s(0)))

  (16)

  1.1.2 Reduction Pair Processor with Usable Rules

  Using the argument filter

  π(p#)

  =

  1

  in combination with the following Weighted Path Order with the following precedence and status

  prec(0#)	=	0		status(0#)	=	[]		list-extension(0#)	=	Lex
  prec(s)	=	1		status(s)	=	[1]		list-extension(s)	=	Lex
  prec(activate)	=	0		status(activate)	=	[]		list-extension(activate)	=	Lex
  prec(activate#)	=	0		status(activate#)	=	[]		list-extension(activate#)	=	Lex
  prec(f)	=	0		status(f)	=	[]		list-extension(f)	=	Lex
  prec(n__s)	=	1		status(n__s)	=	[]		list-extension(n__s)	=	Lex
  prec(p)	=	1		status(p)	=	[]		list-extension(p)	=	Lex
  prec(0)	=	0		status(0)	=	[]		list-extension(0)	=	Lex
  prec(s#)	=	0		status(s#)	=	[]		list-extension(s#)	=	Lex
  prec(n__f)	=	0		status(n__f)	=	[]		list-extension(n__f)	=	Lex
  prec(f#)	=	0		status(f#)	=	[1]		list-extension(f#)	=	Lex
  prec(n__0)	=	0		status(n__0)	=	[]		list-extension(n__0)	=	Lex
  prec(cons)	=	0		status(cons)	=	[]		list-extension(cons)	=	Lex

  and the following Max-polynomial interpretation

  [0#]	=	0
  [s(x1)]	=	x1 + 7721
  [activate(x1)]	=	1
  [activate#(x1)]	=	1
  [f(x1)]	=	1
  [n__s(x1)]	=	1
  [p(x1)]	=	x1 + 0
  [0]	=	1
  [s#(x1)]	=	1
  [n__f(x1)]	=	1
  [f#(x1)]	=	x1 + 1
  [n__0]	=	1
  [cons(x1, x2)]	=	x1 + 1

  together with the usable rules

  p(s(X))	→	X	(3)
  s(X)	→	n__s(X)	(5)
  0	→	n__0	(6)

  (w.r.t. the implicit argument filter of the reduction pair), the pair

  f#(s(0))

  →

  f#(p(s(0)))

  (16)

  could be deleted.

  1.1.2.1 Dependency Graph Processor

  The dependency pairs are split into 0 components.